Classification of graded left-symmetric algebraic structures on Witt and Virasoro algebras

摘要

We find that a compatible graded left-symmetric algebraic structure on the Witt algebra induces an indecomposable module V of the Witt algebra with one-dimensional weight spaces by its left-multiplication operators. From the classification of such modules of the Witt algebra, the compatible graded left-symmetric algebraic structures on the Witt algebra are classified. All of them are simple and they include the examples given by [Comm. Algebra 32 (2004) 243251; J. Nonlinear Math. Phys. 6 (1999) 222245]. Furthermore, we classify the central extensions of these graded left-symmetric algebras which give the compatible graded left-symmetric algebraic structures on the Virasoro algebra. They coincide with the examples given by [J. Nonlinear Math. Phys. 6 (1999) 222245].